Blocking in Fives and Threes: playing to deny a score

In Fives and Threes your move has two effects. It scores for you, and it leaves two open ends for the opponent to play against. Most players only look at the first. This post looks at the second: how to count what a pair of ends allows the opponent to score, and when it is worth taking a smaller score yourself to keep that number low. The scoring rule is in how Fives and Threes scoring works.

What the opponent needs

The opponent scores when they place a tile and the two ends then total a multiple of three or five. Take one end, say a 6, and suppose the other end shows 3. A tile with a 6 on it goes on the 6 end and exposes its other half, and that half plus 3 is the new total. A double exposes both its spots, so 6-6 on the 6 end shows 12, and 12 + 3 = 15, which scores 5 + 3 = 8.

So for any pair of ends you can list every tile that would score and what it would be worth. Two figures matter: how many tiles score, and the best reply among them.

A worked example

The open ends are 1 and 3, total 4, which scores nothing. You hold 1-6 and 3-1, and either can be played.

• Play 1-6 on the 1. The ends become 6 and 3, total 9. Nine divides by three, so you score 9 ÷ 3 = 3.
• Play 3-1 on the 3. The ends become 1 and 1, total 2. It divides by neither three nor five, so you score 0.

Now the replies. With ends 6 and 3, the opponent's scoring tiles are these:

That is seven tiles, and the best pays 8. The 6-3 can be played either way round; its best is on the 3, 6 + 6 = 12, which pays 4 (on the 6 it would show 3, and 3 + 3 = 6 pays only 2).

With ends 1 and 1, only tiles carrying a 1 can be played, and the scoring ones are these:

• 1-1 (double shows 2): 2 + 1 = 3, scores 1.
• 1-2: 2 + 1 = 3, scores 1.
• 1-4: 4 + 1 = 5, scores 1.
• 1-5: 5 + 1 = 6, scores 6 ÷ 3 = 2.

The other tiles with a 1 fail: 1-0 gives 1, 1-3 gives 4 and 1-6 gives 7. So there are four scoring tiles and the best pays 2. The 1-6 and 1-3 are out of reach in any case: you hold one and the other is in the line.

Comparing the two plays

Play 1-6 and you take 3 now, but the opponent may hold the 6-6 and answer with 8. In the worst case that is 3 − 8 = −5 over the exchange. Play 3-1 and you take nothing, but the best reply is 2, so the worst case is 0 − 2 = −2. The worst case favours the quiet play by 3 points.

Your own hand and the count

The 8 depends entirely on one tile, the 6-6. If you hold it, nobody else can play it. Remove it from the list and the ends 6 and 3 leave six scoring tiles, and the best reply drops to 4 (the 6-3 and 3-3, both 12). Then the comparison reads 3 − 4 = −1 for the scoring play against −2 for the quiet one, and taking the 3 is right. This is the same subtraction as in counting tiles: tiles you hold, or can see in the line, cannot be played against you.

The doubles do most of the damage because they expose two spots at once, as covered in doubles at an open end. Any total above 12 needs one, so an end that matches a double you hold, or a double already seen, is a safer end.

Some quiet pairs of ends

For reference, here are pairs of plain ends and what they allow, counted the same way across the full set of 28 tiles:

Both the 5-and-5 and 6-and-3 pairs can allow 15. With 5 and 5, the 5-5 double shows 10 and 10 + 5 = 15. A short list of scoring tiles does not make a pair safe if one of them pays 8, and a pair like 3 and 3 is gentler, with a best reply of 3. The full list of what each total pays is in every scoring total, 3 to 20.

When not to block

Denying a score costs you something whenever the quiet tile scores less than the loud one, so do the sum rather than defend by habit. Taking the 15 yourself, as in why 15 is the best total, is almost always worth the risk. Block when the two plays score about the same for you, or when the reply you are removing is a large one.